本文關鍵詞： 美式期權 Black-scholes模型 自由邊界 有限差分法 二叉樹方法 LSM　出處：《山東大學》2012年碩士論文　論文類型：學位論文

【摘要】：本文第一章是緒論部分,簡要介紹了期權的定義與分類情況以及期權定價的理論基礎,其中主要介紹了風險中性原理、無套利定價原理和期權的價值分析,在這一章的最后對期權定價理論從無到有,從簡到繁的發(fā)展過程做了一番介紹,一代一代的專家學者都為此作出了卓越的貢獻。 由于Black與Scholes作出的尤其突出的貢獻,我們在第二章介紹了Black-Scholes模型的前提假設、建立過程以及定價公式的推導過程,重溫了Black與Scholes的工作,另一方面,我們也發(fā)現(xiàn)了Black-Scholes模型的缺點：只適用于歐式期權,但是我們本文研究的是美式期權,即便如此,我們的工作并不是一無是處,在接下來的第三章我們重點介紹的是美式期權的定價模型,首先明確說明美式期權的定價問題屬于自由邊界問題,根據(jù)美式期權的特點我們建立了兩種形式的定價模型：拋物型方程模型和變分不等式方程模型。在本章的最后一節(jié)我們又推出了美式期權定價的看漲—看跌對稱關系。 光有模型不行,還需要求解方法,為此,在接下來的第四、五、六章分別介紹了美式期權的三種解法：有限差分法、二叉樹方法、最小二乘蒙特卡洛模擬方法。 有限差分法是先介紹了差分方法,然后基于變分不等式模型的離散化進行了推導,得出了顯示差分格式和隱式差分格式,其中顯示差分格式還給出了Matlab程序,以及一個數(shù)值算例,最后給出了差分方法的評價。 二叉樹方法那一章先介紹了二叉樹方法的定義,然后給出了二叉樹的定價過程,再根據(jù)美式期權的特點給出了美式期權的定價過程,最后給出了方法評價,這一章屬于介紹性質(zhì)的內(nèi)容,原因是二叉樹方法中參數(shù)的選擇不確定,在以往的研究中有幾位優(yōu)秀的專家學者分別給出了不一樣的參數(shù)選取方法,只能具體情況具體分析了,再加上二叉樹方法的“維數(shù)效應”問題,所以應用很有限,本文只做這些介紹而已。 最小二乘蒙特卡洛方法是最近出現(xiàn)的新方法,不是我的原創(chuàng),但是很新穎,應用起來也很方便,很廣泛就把它也加入進來了,詳細地介紹之后,自己進行了Matlab編程,也進行了數(shù)值實驗,最后給出了方法評價。 做了這么多工作之后發(fā)現(xiàn),本篇論文的結(jié)構是：期權介紹,B-S模型,美式期權模型的建立,美式期權定價的求解方法。
[Abstract]:The first chapter is the introduction, which briefly introduces the definition and classification of options and the theoretical basis of option pricing, including risk neutral principle, no-arbitrage pricing principle and option value analysis. At the end of this chapter, the theory of option pricing has been introduced from scratch, from simplicity to complexity, and a generation of experts and scholars have made outstanding contributions to it. Because of the outstanding contribution made by Black and Scholes, in the second chapter, we introduce the premise hypothesis of Black-Scholes model, the establishment process and the derivation process of pricing formula, review the work of Black and Scholes, on the other hand, We also found the disadvantage of the Black-Scholes model: only for European options, but we are studying American options in this paper, but even so, our work is not useless. In the following chapter, we focus on the pricing model of American option. Firstly, we clearly explain that the pricing problem of American option belongs to the free boundary problem. According to the characteristics of American option, we establish two kinds of pricing models: parabolic equation model and variational inequality equation model. In the last section of this chapter, we derive the bullish-bear-bear-symmetry relation of American option pricing. For this reason, in the following chapters 4th, 5 and 6, three solutions of American option are introduced: finite difference method, binary tree method and least square Monte Carlo simulation method. The finite difference method is introduced first, then based on the discretization of variational inequality model, the display difference scheme and implicit difference scheme are derived. The display difference scheme and the Matlab program are also given. And a numerical example is given. Finally, the evaluation of the difference method is given. In the chapter of binary tree method, the definition of binary tree method is introduced, then the pricing process of binary tree is given, and then the pricing process of American option is given according to the characteristics of American option. Finally, the method evaluation is given. This chapter belongs to the content of introduction, the reason is that the choice of parameters in binary tree method is uncertain. In the past research, several excellent experts and scholars have given different parameter selection methods, which can only be analyzed concretely. In addition, due to the "dimension effect" problem of binary tree method, the application is very limited. The least square Monte Carlo method is a new method that has recently emerged, not my original one, but it is very novel and convenient to use. It is also widely included. After a detailed introduction, I have carried out Matlab programming myself. Numerical experiments are also carried out, and the evaluation method is given. After doing so much work, it is found that the structure of this paper is: option introduction to B-S model, the establishment of American option model, the solution of American option pricing.
【學位授予單位】：山東大學
【學位級別】：碩士
【學位授予年份】：2012
【分類號】：F224;F830.9

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本文編號：1550557